A vapor$-$compression refrigeration system operates with the cascade arrangement. Refrigerant $22$ is the working fluid in the high$-$temperature cycle and Refrigerant $134$a is used in the low$-$temperature cycle. For the Refrigerant $134$a cycle, the working fluid enters the compressor as saturated vapor at $-30F$ and is compressed isentropically to $50lb\frac{f}{i}{n}^2.$ Saturated liquid leaves the intermediate heat exchanger at $50lb\frac{f}{i}{n}^2$ and enters the expansion valve. For the Refrigerant $22$ cycle, the working fluid enters the compressor as saturated vapor at a temperature $5F$ below that of the condensing temperature of the Refrigerant $134$a in the intermediate heat exchanger. The Refrigerant $22$ is compressed isentropically to $250lb\frac{f}{i}{n}^2.$ Saturated liquid then enters the expansion valve at $250lb\frac{f}{i}{n}^2.$ The refrigerating capacity of the cascade system is $22$ tons. Determine

$($a$)$ The power input to each compressor $($Btu$/$min$)$

$($b$)$ The overall coefficient of performance of the cascade cycle $($no units$)$